document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=610698>Question 610698</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the product of two numbers is 84. One number is 13 more than 2 times the other. Find both numbers </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #384515 by </B><A HREF=/tutors/aboutme.mpl?userid=Maths68><B>Maths68(1474)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Maths68><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Maths68><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=384515\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x and y be the two numbers\r<BR>\n" );
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document.write( "The product of two numbers is 84.<BR>\n" );
document.write( "xy=84.........(1)<BR>\n" );
document.write( "One number is 13 more than 2 times the other.<BR>\n" );
document.write( "x=2y+13..............(2)<BR>\n" );
document.write( "Substitute the value of x in (1)<BR>\n" );
document.write( "(2y+13)y=84<BR>\n" );
document.write( "2y^2+13y=84<BR>\n" );
document.write( "2y^2+13y-84=0<BR>\n" );
document.write( "Solve for y<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ay%5E2%2Bby%2Bc=0 ALIGN=MIDDLE  ALT=\"ay%5E2%2Bby%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%5E2%2B13y%2B-84+=+0 ALIGN=MIDDLE  ALT=\"2y%5E2%2B13y%2B-84+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%2813%29%5E2-4%2A2%2A-84=841 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2813%29%5E2-4%2A2%2A-84=841\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=841 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-13%2B-sqrt%28+841+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-13%2B-sqrt%28+841+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+%28-%2813%29%2Bsqrt%28+841+%29%29%2F2%5C2+=+4 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+%28-%2813%29%2Bsqrt%28+841+%29%29%2F2%5C2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+%28-%2813%29-sqrt%28+841+%29%29%2F2%5C2+=+-10.5 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+%28-%2813%29-sqrt%28+841+%29%29%2F2%5C2+=+-10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%5E2%2B13y%2B-84 ALIGN=MIDDLE  ALT=\"2y%5E2%2B13y%2B-84\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%5E2%2B13y%2B-84+=+2%28y-4%29%2A%28y--10.5%29 ALIGN=MIDDLE  ALT=\"2y%5E2%2B13y%2B-84+=+2%28y-4%29%2A%28y--10.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 4, -10.5.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B13%2Ax%2B-84+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B13%2Ax%2B-84+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "So<BR>\n" );
document.write( "y=4 or y=-10.5(inadmissible) <BR>\n" );
document.write( "Put the value of y=4 in (1)<BR>\n" );
document.write( "xy=84<BR>\n" );
document.write( "4x=84<BR>\n" );
document.write( "x=84/4<BR>\n" );
document.write( "x=21\r<BR>\n" );
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document.write( "x,y = 21,4\r<BR>\n" );
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document.write( "Check<BR>\n" );
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document.write( "xy=84<BR>\n" );
document.write( "21*4=84<BR>\n" );
document.write( "84=84\r<BR>\n" );
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document.write( "x=2y+13<BR>\n" );
document.write( "21=2(4)+13<BR>\n" );
document.write( "21=8+13<BR>\n" );
document.write( "21=21<BR>\n" );
document.write( " </SPAN>\n" );
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